Quadratic Functions and their Characteristics Unit 6 Quadratic Functions Math II

Mini Lesson: Domain and Range Recall that a set of ordered pairs is also called a relation. The domain is the set of x-coordinates of the ordered pairs. The range is the set of y-coordinates of the ordered pairs. Interval Notation is the way to represent the domain and range of a function as an interval pair of numbers. – Examples: [-2, 3], (0, 2], (-∞, ∞) – The numbers are the end points of the interval – Use parenthesis if the endpoint is NOT included – Use brackets if the endpoint IS included – ∞ (Infinity) : Use this if numbers go on forever in the positive direction – -∞ (Negative Infinity) : Use this if numbers go on forever in the negative direction

Find the domain and range of the function graphed to the right. Use interval notation. Domain: [ -3, 4 ] Range: [ -4, 2 ] x y Introduction to Domain and Range Domain and Range from Graphs

Introduction to Domain and Range Find the domain and range of the function graphed to the right. Use interval notation. x y Domain: ( - ∞, ∞ ) Range: [ -2, ∞ ) Domain and Range from Graphs

Introduction to Domain and Range Write the domain and range in interval notation. Domain: Range:

Introduction to Domain and Range Write the domain and range in interval notation. Domain: Range:

Introduction to Domain and Range Write the domain and range in interval notation. Domain: Range:

Introduction to Domain and Range Write the domain and range in interval notation. Domain: Range:

Introduction to Domain and Range Write the domain and range in interval notation. Domain: Range:

Introduction to Domain and Range Write the domain and range in interval notation. Domain: Range:

Introduction to Domain and Range Write the domain and range in interval notation. Domain: Range:

Introduction to Domain and Range Write the domain and range in interval notation. Domain: Range:

Introduction to Domain and Range Domain and Range worksheet! Find the Domain and Range of each graph. Whoever can finish 1 st, 2 nd, and 3 rd with all questions correct will get a piece of candy!

Introduction to Quadratic Functions A quadratic function always has a degree of 2. – This means there will always be an x 2 in the equation and never any higher power of x. The shape of a quadratic function’s graph is called a parabola. -It looks like a “U” The Standard Form of a Quadratic is y = a x 2 + b x + c Where a, b, and c can be real numbers with a ≠ 0.

Introduction to Quadratic Functions Vertex Axis of Symmetry Domain: (-∞,∞∞) Range: [0, ∞) Most Basic Quadratic Function: y = x 2

Introduction to Quadratic Functions X-Intercepts Roots Zeroes Solutions

Quadratic vocabulary - Axis of Symmetry – the line that divides a parabola into 2 parts that are mirror images. The axis of symmetry is always a vertical line defined by the x-coordinate of the vertex. (ex: x = 0) - Vertex – the point at where the parabola intersects that axis of symmetry. The y-value of the parabola represents the maximum or minimum value of the function. - X-Intercepts, Zeroes, Roots, Solutions – All of these terms mean the same thing and refer to where the parabola crosses the x-axis. When asked to solve a quadratic, this is what we are looking for! - Minimum – If the graph opens up (smiles) and the vertex is the lowest point on the graph, then the vertex is a minimum. - Maximum - If the graph open down (frowns) and the vertex is the highest point on the graph, then the vertex is a maximum.

Example: Find the vertex, axis of symmetry, zeroes, and domain and range of the quadratic function. Determine if the vertex is a minimum or a maximum. Vertex: Axis of Symmetry: Zeroes: Domain: Range: Min or Max?: Vertex: Axis of Symmetry: Zeroes: Domain: Range: Min or Max?:

Back to Standard Form ! y = a x 2 + b x + c -If a > 0, then vertex will be a minimum. -If a < 0, then vertex will be a maximum. -If in standard form, use the formula to find axis of symmetry. – This will also be the x coordinate of the vertex, substitute that into the original equation to find the y coordinate.

Example: Put each in Standard Form. Determine whether each function is a quadratic. 1.f(x) = (-5x – 4)(-5x – 4) 1.y = 3(x – 1) y = x – 11x – x 2 1.f(x) = 3x(x + 1) – x 1.y = 2(x + 2) 2 – 2x 2

Example: Determine whether the vertex will be a maximum or minimum. Find the Axis of Symmetry and Vertex of each. 1.y = x 2 – 4x y = -3x 2 + 6x y = 2x 2 – 8x y = -x 2 – 8x – 15

Finding Quadratic Models! (in your calculator) Find a quadratic model for a set of values. - Step 1 : Enter the data into the calculator (STAT -> Edit) (x’s in L1, y’s in L2) - Step 2 : Calculate the Quadratic Regression model by hitting STAT again  Calc, then 5: QuadReg - Step 3 : Substitute the given a, b, and c values into the standard form.

Example: Find a quadratic model for each set of values. 1.(1, -2), (2, -2), (3, -4) 2. x 13 f(x) 38

Quadratic Model Application A man throws a ball off the top of a building. The table shows the height of the ball at different times. a.Find a quadratic model for the data. b.Use the model to estimate the height of the ball at 2.5 seconds. Height of a Ball TimeHeight 0 s46 ft 1 s63 ft 2 s48 ft 3 s1 ft